Call your local pound. He may be a dumped cat. If he does belong to someone, he's definitely being ignored.

You can call your local Humane Society and ask for advice. To really help him you'll need to try to get him comfortable being around you (canned food may work) so that you can capture him. Then take him to a vet or to your local no-kill shelter.
Also call any cat rescue groups in your area and see if they can help.

If he's so wary of people that you can't entice him with food, then I doubt he belongs to anyone. Other than posting photos of the cat around your neighborhood, along with a contact phone number, I don't think there's any way to find out where he lives.

I once had my doubts about whether a beautiful Siamese cat that came to my door often was a stray or not. Not knowing what to do, I managed to catch it and put my old cat's collar and tags on it, with my cat's name and my phone number. I figured if the cat belonged to a neighbor, the neighbor would at least take the collar off if not call me. When the cat was still wearing the collar a few days later I rescued it.

You could trap the cat and take him to the vet. Then you could either keep him or turn him loose again after his foot is healed, but either way he would not be suffering.

He might have an owner, or he might not. Some feral cats who are being fed can appear quite healthy. I once had a cat come through my pet door, decided to keep him, took him to my vet for innoculations and a check up, then discovered he was leading a double life and my neighbors across the street had been feeding and looking after him for some time, and he'd had the same round of vaccinations at their own vet. He may have had another home before he showed up at our houses.

Yeah, I will probably have to get him coming around more by feeding him. I can't trap him as I will probably trap my own cats and several others before catching him! My next question is if anyone knows if vets are willing to work with situations like this. In other words, I'm not in a position to pay for his healing, at least not all at one time. If they would let me pay a little at a time since he's not my cat, I might be able to swing it.

aviolet, thank you for helping the poor little kitty! I would call around to vets in your area as well as rescue/humane organizations and ask your question. No doubt some such places may reduce the cost or certainly let you pay over time given the situation. Please report back and let us know what happens.

I will bet this cat does not have a broken foot, but an abscessed foot from being bitten by another cat... pretty common (and broken feet in cats is rare, though of course it can happen)... that cat will need antibiotics, hopefully pain meds, and probably be OK after.. .needs to get neutered, though!

He's going to be hard to catch because he's getting enough food other places that he's not starved, and he won't let me get very close. But I think he is a stray. I can't get a good look at his leg/foot, but he sure doesn't use it any.

Any update?

If he is someone's cat, that person doesn't deserve to have a pet. If someone is neglecting their cat to the point that they let it outside with an injured foot, they have no right at that point to claim ownership over that cat IMHO.

This is a perfect example of why cats should not be allowed outdoors. Once a cat is outdoors, it is fair game. For getting hit by a car, injured, adopted by someone else, or any other number of things.

If you are concerned about the cat, do what you can to trap it some how and get it some medical treatment.

We recently adopted a very sweet and friendly cat that lived on our street. It was outside a lot and was unneutered. It would run across our side street every day to see us, and it would sometimes sit at the corner house looking in the door. It may have been theirs - I don't care. As far as I'm concerned, if they make the cat stay out in the rain while it's looking in on the warm house, don't neuter it, and only want to deal with him when it's convenient, it's not their pet to claim ownership of.

After having him a few months, he developed a bad UTI. It may have been there the whole time we had him and flaired up over time with the adjustments. This may be why the people on the corner made it stay outside. They could have simply taken a trip to the vet to get antibiotics. People are cruel.

The last thing that should be on your mind with this cat is who owns it.

I am convinced by now that the cat does not belong to anyone anymore, though it probably did at one time. It does cover a good distance, even on 3 legs. It visits certain yards but never stays too long in any of them. I saw it drinking from another cat's water dish - I am sure it gets its food the same way. I haven't seen it near my house as much and it runs if I try to approach it, even with food. I think it has become more wary the longer it's been homeless, and injured. I want to help this cat but I may not be able to. At least it seems to have found enough resources on its own and it gets around.

I suggest not approaching the cat but to leave food and water for it, especially when you know it's in your yard. It'll start to hang out more at your place if he knows he can find food there and he may become more comfortable around you.
Start by leaving food in the area of the yard you see him in and then gradually move the food closer to the house.
Will he hang around if you're in the yard, and ignoring him, or does he run when he sees you?